Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

Ahmed Elshenhab; Xingtao Wang; Fatemah Mofarreh; Omar Bazighifan

doi:10.32604/cmes.2022.021512

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ARTICLE

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

Ahmed M. Elshenhab^1,2,*, Xingtao Wang¹, Fatemah Mofarreh³, Omar Bazighifan^4,*

1 School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
3 Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, 11546, Saudi Arabia
4 Section of Mathematics, International Telematic University Uninettuno, Roma, 00186, Italy

* Corresponding Authors: Ahmed M. Elshenhab. Email: email ; Omar Bazighifan. Email: email

Computer Modeling in Engineering & Sciences 2023, 134(2), 927-940. https://doi.org/10.32604/cmes.2022.021512

Received 18 January 2022; Accepted 23 March 2022; Issue published 31 August 2022

Abstract

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay. By using new conformable delayed matrix functions and the method of variation, we obtain a representation of their solutions. As an application, we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions. The obtained results are new, and they extend and improve some existing ones. Finally, an example is presented to illustrate the validity of our theoretical results.

Keywords

Representation of solutions; conformable fractional derivative; conformable delayed matrix function; conformable fractional delay differential equations; finite time stability

1 Introduction

In recent years, particularly in 2014, Khalil et al. [1] introduced a new definition of the fractional derivative called the conformable fractional derivative that extends the classical limit definition of the derivative of a function. The conformable fractional derivative has main advantages compared with other previous definitions. It can, for example, be used to solve the differential equations and systems exactly and numerically easily and efficiently, it satisfies the product rule and quotient rule, it has results similar to known theorems in classical calculus, and applications for conformable differential equations in a variety of fields have been extensively studied, see [2–10] and the references therein. On the other hand, in 2003, Khusainov et al. [11] represented the solutions of linear delay differential equations by constructing a new concept of a delayed exponential matrix function. In 2008, Khusainov et al. [12] adopted this approach to represent the solutions of an oscillating system with pure delay by establishing a delayed matrix sine and a delayed matrix cosine. This pioneering research yielded plenty of novel results on the representation of solutions, which are applied in the stability analysis and control problems of time-delay systems; see for example [13–28] and the references therein. Thereafter, in 2021, Xiao et al. [29] obtained the exact solutions of linear conformable fractional delay differential equations of order $α∈(0,1]$ by constructing a new conformable delayed exponential matrix function.

However, to the best of our knowledge, no study exists dealing with the representation and stability of solutions of conformable fractional delay differential systems of order $α∈(1,2]$ .

Motivated by these papers, we consider the explicit formula of solutions of linear conformable fractional differential equations with pure delay

$(D0αy)(x)=−By(x−τ)+f(x),for x≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for - τ≤x≤0,$ (1)

by constructing new conformable delayed matrix functions. Moreover, the representation of solutions of Eq. (1) is used to obtain a finite time stability result on $W=[0,L]$ , $L>0$ , where $D0α$ is called the conformable fractional derivative of order $α∈(1,2]$ with lower index zero, $y(x)∈Rn$ , $ψ∈C2([−τ,0],Rn)$ , $B∈Rn×n$ is a constant nonzero matrix and $f∈C([0,∞),Rn)$ is a given function.

The paper is organized as follows: In Section 2, we present some basic definitions concerning conformable fractional derivative and finite time stability, and construct new conformable delayed matrix functions and derive their properties for use when we discuss the representation of solutions and finite time stability. In Section 3, by using the new conformable delayed matrix functions, we give the explicit formula of solutions of Eq. (1). In Section 4, as an application, we derive a finite time stability result using the representation of solutions. Finally, we give an example to illustrate the main results.

2 Preliminaries

Throughout the paper, we denote the vector norm and matrix norm, respectively, as $‖y‖=∑i=1n|yi|$ and $‖B‖=max1≤j≤n∑i=1n|bij|$ ; $yi$ and $bij$ are the elements of the vector y and the matrix B, respectively. Denote $C(W,Rn)$ the Banach space of vector-value continuous function from $W→Rn$ endowed with the norm $‖y‖C=maxx∈W‖y(x)‖$ for a norm $‖⋅‖$ on $Rn$ . We introduce a space $C1(W,Rn)={y∈C(W,Rn):y′∈C(W,Rn)}$ . Furthermore, we see $‖ψ‖C=maxυ∈[−τ,0]‖ψ(υ)‖$ .

We recall some basic definitions of conformable fractional derivative, fractional exponential function, and finite time stability.

Definition 2.1. ([2, Definition 2.2]). Let $f:[a,∞)→Rn$ be a differentiable function at x. Then the conformable fractional derivative for f of order $α=(1,2]$ is given by

$Daα(f)(x)=limε→0⁡f′(x+ε(x−a)2−α)−f′(x)ε,x>a,$

if the limit exists.

Remark 2.1. As a consequence of Definition 2.1, we can show that

$Daα(f)(x)=(x−a)2−αf′′(x),$

where $α=(1,2]$ , and f is $2$ -differentiable at $x>a$ .

Definition 2.2. ([2]). We define the fractional exponential function as follows:

$Eα(λ,x−a)=exp⁡(λ.(x−a)αα)=∑k=0∞λk(x−a)αkαkk!,α>0,λ∈R.$

Definition 2.3. ([30]). The system in Eq. (1) is finite time stable with respect to ${0,W,τ,δ,β}$ , $δ<β$ if and only if $η<δ$ implies $‖y(x)‖<β$ for all $x∈W$ , where $η=max{‖ψ‖C,‖ψ′‖C,‖ψ′′‖C}$ and $δ$ , $β$ are real positive numbers.

Next, we construct new conformable delayed matrix functions that are the fundamental solution matrices of Eq. (1).

Definition 2.4. The conformable delayed matrix functions $Hτ,α(Bxα)$ and $Mτ,α(Bxα)$ are defined as

$Hτ,α(Bxα):={Θ,−∞<x<−τ,I,−τ≤x<0,I−B1α(α−1)xα,0≤x<τ,⋮⋮I−B1α(α−1)xα+B212!α2(α−1)(2α−1)(x−τ)2α+⋯+(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα,(m - 1)τ≤x<mτ,$ (2)

$Mτ,α(Bxα):={Θ,−∞<x<−τ,I(x+τ),−τ≤x<0,I(x+τ)−B1α(α+1)xα+10≤x<τ,⋮⋮I(x+τ)−B1α(α+1)xα+1+B212!α2(α+1)(2α+1)(x−τ)2α+1+⋯+(−1)mBm1m!αm∏i=1m(iα+1)(x−(m−1)τ)mα+1,(m - 1)τ≤x<mτ,$ (3)

respectively, where $m=0,1,2,…$ , I is the $n×n$ identity matrix and $Θ$ is the $n×n$ null matrix.

Lemma 2.1. The following rule is true:

$D0αHh,α(Bxα)=−BHh,α(B(x−h)α).$

Proof. First, when $x∈(−∞,−τ)$ , we obtain $Hτ,α(Bxα)=Hτ,α(B(x−τ)α)=Θ$ , and we can see that Lemma 2.1 holds. Following that, set $(m−1)τ≤x<mτ$ , $m=0,1,2,…$ , we get

$Hτ,α(Bxα)=I−B1α(α−1)xα+B212!α2(α−1)(2α−1)(x−τ)2α+⋯+(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα.$

Applying Remark 2.1, we get

$D0αHτ,α(Bxα)=D0αI−D0α[B1α(α−1)xα]+Dτα[B212!α2(α−1)(2α−1)(x−τ)2α]+⋯+D(m−1)τα[(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα]=Θ−B+B21α(α−1)(x−τ)α−B312!α2(α−1)(2α−1)(x−2τ)2α+⋯+(−1)mBm1(m−1)!αm−1∏i=1m−1(iα−1)(x−(m−1)τ)(m−1)α=−B[I−B1α(α−1)(x−τ)α+B212!α2(α−1)(2α−1)(x−2τ)2α+⋯+(−1)m−1Bm−11(m−1)!αm−1∏i=1m−1(iα−1)(x−(m−1)τ)(m−1)α]=−BHτ,α(B(x−τ)α).$

This completes the proof.

In the same way that we proved Lemma 2.1, we can derive the next result.

Lemma 2.2. The following rule is true:

$D0αMh,α(Bxα)=−BMh,α(B(x−h)α).$

To conclude this section, we provide a norm estimation of the conformable delayed matrix functions, which is used while discussing finite time stability.

Lemma 2.3. For any $x∈[(m−1)τ,mτ]$ , $m=0,1,2,…$ , we have

$‖Hτ,α(Bxα)‖≤Eα(‖B‖α−1,x).$

Proof. Taking the norm of Eq. (2), we get

$‖Hτ,α(Bxα)‖≤1+‖B‖xαα(α−1)+‖B‖2(x−τ)2α2!α2(α−1)(2α−1)+⋯+‖B‖m(x−(m−1)τ)mαm!αm∏i=1m(iα−1)≤1+‖B‖xαα(α−1)+‖B‖2x2α2!α2(α−1)2+⋯+‖B‖mxmαm!αm(α−1)m≤∑k=0∞‖B‖kxαk(α−1)kαkk!=Eα(‖B‖α−1,x).$

This completes the proof.

Lemma 2.4. For any $x∈[(m−1)τ,mτ]$ , $m=0,1,2,…$ , we have

$‖Mτ,α(Bxα)‖≤(x+τ)Eα(‖B‖α+1,x+τ).$

Proof. Taking the norm of Eq. (3), we get

$‖Mτ,α(Bxα)‖≤(x+τ)+‖B‖1α(α+1)xα+1+‖B‖212!α2(α+1)(2α+1)(x−τ)2α+1+⋯+‖B‖m1m!αm∏i=1m(iα+1)(x−(m−1)τ)mα+1≤(x+τ)+‖B‖(x+τ)α+1α(α+1)+‖B‖2(x+τ)2α+12α2(α+1)2+⋯+‖B‖m(x+τ)mα+1m!αm(α+1)m≤∑k=0∞‖B‖k(x+τ)kα+1k!αk(α+1)k=(x+τ)Eα(‖B‖α+1,x+τ).$

This completes the proof.

3 Exact Solutions for Linear Conformable Fractional Delay Systems

In this section, we give the exact solutions of Eq. (1) via the conformable delayed matrix functions and the method of variation of constants. To do this, we consider the homogeneous system of linear conformable fractional delay differential equations

$(D0αy)(x)=−By(x−τ),forx≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for−τ≤x≤0,$ (4)

and the linear inhomogeneous conformable fractional delay system

$(D0αy)(x)=−By(x−τ)+f(x),forx≥0,τ>0,y(x)≡Θ,y′(x)≡Θfor−τ≤x≤0.$ (5)

Theorem 3.1. The solution $y(x)$ of Eq. (4) has the representation

$y(x)={ψ(x),−τ≤x≤0,Hτ,α(Bxα)ψ(−τ)+Mτ,α(Bxα)ψ′(−τ)+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αψ(υ)dυ,x≥0.$ (6)

Proof. We seek for a solution of Eq. (4) in the form

$y(x)=Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αr(υ)dυ,$ (7)

or

$y(x)=Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ,$

where $c1$ and $c2$ are unknown constants vectors on $Rn$ , and $r(x)$ is an unkown twice continuously differentible vector function. From Lemmas 2.1 and 2.2, we deduce that $Hτ,α(Bxα)$ and $Mτ,α(Bxα)$ are solutions of Eq. (4). We notice that Eq. (6) is a solution of Eq. (4) due to the linearity of solutions for arbitrary $c1$ , $c2$ and $r(x)$ . Now we find the constants $c1$ and $c2$ , and the vector function $r(x)$ so that the initial conditions $y(x)≡ψ(x)$ , $y(x)≡ψ′(x)$ for $−τ≤x≤0$ , are satisfied. That is, the following relations hold for $−τ≤x≤0$ :

$Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ=ψ(x),$ (8)

and

$ddx{Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ}=ψ′(x).$ (9)

Consider Eq. (8). If $−τ≤x<0$ , then

$Hτ,α(Bxα)=I,Mτ,α(Bxα)=I(x + τ),$

and

$Mτ,α(B(x−τ−υ)α)={I(x−υ),υ∈[−τ,x],Θ,υ∈(x,0],$

which implies that

$c1+(x+τ)c2+∫−τx(x−υ)r′′(υ)dυ=ψ(x),$ (10)

and

$∫−τx(x−υ)r′′(υ)dυ=−(x+τ)r′(−τ)+r(x)−r(−τ).$ (11)

Substituting Eq. (11) into Eq. (10), we get

$(c1−r(−τ))+(c2−r′(−τ))(x+τ)+(r(x)−ψ(x))=Θ.$ (12)

Differentiating Eq. (12) with respect to x, we have

$(c2−r′(−τ))+(r′(x)−ψ′(x))=Θ.$ (13)

As a result, we find that the equalities obtained Eqs. (12) and (13) are true if

$c1=ψ(−τ),c2=ψ′(−τ),r(x)=ψ(x).$ (14)

Substituting Eq. (14) into Eq. (7), we obtain Eq. (6). This finishes the proof.

Theorem 3.2. The particular solution $y0(x)$ of Eq. (5) has the representation

$y0(x)=∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ.$ (15)

Proof. We try to find a particular solution $y0(x)$ of Eq. (5) in the form

$y0(x)=∫0xMτ,α(B(x−τ−υ)α)ξ(υ)dυ,$ (16)

by applying the method of variation of constants, where $ξ(υ)$ , $0<υ≤x$ , is an unknown function. Taking the conformable derivative of Eq. (16), we get

$D0αy0(x)=∫0xD0αMτ,α(B(x−τ−s)α)ξ(υ)dυ+x2−αξ(x)=−B∫0xMτ,α(B(x−2τ−υ)α)ξ(υ)dυ+x2−αξ(x).$ (17)

Substituting Eqs. (16) and (17) into Eq. (5), and noting that

$∫x−τxMτ,α(B(x−2τ−υ)α)ξ(υ)dυ=Θ,$

We have $x2−αξ(x)=f(x)$ . Substituting $ξ(x)=xα−2f(x)$ into Eq. (16), we obtain Eq. (15). This completes the proof.

Corollary 3.1. The solution $y(x)$ of Eq. (1) can be represented as

$y(x)={ψ(x),−τ≤x≤0,Hτ,α(Bxα)ψ(−τ)+Mτ,α(Bxα)ψ′(−τ)+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αψ(υ)dυ+∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ,x≥0.$ (18)

Remark 3.1. Let $α=2$ in Eq. (1). Then Corollary 3.1 coincides with Corollary 1 in [13].

Remark 3.2. Let $α=2$ , $B=B2$ in Eq. (1) such that the matrix B is a nonsingular $n×n$ matrix. Then

$Hτ,2(B2x2)=cosτ(Bx),Mτ,2(B2x2)=B−1sinτ(Bx).$

where $cosτ(Bx)$ and $sinτ(Bx)$ are called the delayed matrix of cosine and sine type, respectively, defined in [12]. Therefore, Corollary 3.1 coincides with Theorems 1 and 2 in [12].

4 Finite Time Stability of Linear Conformable Fractional Delay Systems

In this section, we establish some sufficient conditions for the finite time stability results of Eq. (1) by using a norm estimation of the conformable delayed matrix functions and the formula of general solutions of Eq. (1).

Theorem 4.1. The system Eq. (1) is finite time stable with respect to ${0,W,τ,δ,β}$ , $δ<β$ if

$Eα(‖B‖α+1,L+τ)<β−δEα(‖B‖α−1,L)−‖f‖Cα(α−1)LαEα(‖B‖α+1,L)δ(L+τ)(τ+1).$ (19)

Proof. By using Definition 2.3, and Theorems 3.1 and 3.2, we have $η<δ$ and

$‖y(x)‖≤‖Hτ,α(Bxα)‖‖ψ(−τ)‖+‖Mτ,α(Bxα)‖‖ψ′(−τ)‖+‖∫−τ0Mτ,α(B(x−τ−υ)α)ψ′′(υ)dυ‖+‖∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ‖≤δ‖Hτ,α(Bxα)‖+δ‖Mτ,α(Bxα)‖+δ∫−τ0‖Mτ,α(B(x−τ−υ)α)‖dυ+‖f‖C∫0x‖Mτ,α(B(x−τ−υ)α)‖υα−2dυ.$ (20)

Note that $Mτ,α(Bxα)=Θ$ if $x∈(−∞,−τ)$ . For $−τ≤υ≤0$ , we get

$Mτ,α(B(x−τ−υ)α)={Mτ,α(B(x−τ−υ)α),υ∈[−τ,x],Θ,υ∈(x,0].$

Thus

$‖Mτ,α(B(x−τ−υ)α)‖={‖Mτ,α(B(x−τ−υ)α)‖,υ∈[−τ,x],0,υ∈(x,0].$

Therefore, from Lemma 2.4, we have

$‖Mτ,α(B(x−τ−υ)α)‖≤(x−υ)Eα(‖B‖α+1,x−υ)≤(x+τ)Eα(‖B‖α+1,x+τ),$ (21)

for $−τ≤υ≤0$ , $x∈W$ , and since $Eα(‖B‖α+1,x−υ)$ is increasing function when $x≥υ$ . From Eq. (21), we get

$∫−τ0‖Mτ,α(B(x−τ−υ)α)‖dυ≤τ(x+τ)Eα(‖B‖α+1,x+τ).$ (22)

From Lemma 2.4, we have

$∫0x‖Mτ,α(B(x−τ−υ)α)‖υα−2dυ≤∫0x(x−υ)Eα(‖B‖α+1,x−υ)υα−2dυ≤Eα(‖B‖α+1,x)∫0x(x−υ)υα−2dυ=xαα(α−1)Eα(‖B‖α+1,x).$ (23)

From Eqs. (20), (22) and (23), we get

$‖y(x)‖≤δEα(‖B‖α−1,x)+δ(x+τ)Eα(‖B‖α+1,x+τ)+δτ(x+τ)Eα(‖B‖α+1,x+τ)+‖f‖Cα(α−1)xαEα(‖B‖α+1,x),$ (24)

for all $x∈W$ . Combining Eq. (19) with Eq. (24), we obtain $‖y(x)‖<β$ for all $x∈W$ . This completes the proof.

Corollary 4.1. Let $α=2$ in Eq. (1). Then the system

$y′′(x)=−By(x−τ)+f(x),forx≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for−τ≤x≤0,$

is finite time stable with respect to ${0,W,τ,δ,β}$ , $δ<β$ if

$E2(‖B‖3,L+τ)<β−δE2(‖B‖,L)−‖f‖C2L2E2(‖B‖3,L)δ(L+τ)(τ+1).$

Remark 4.1. Let $α=2$ , $B=B2$ in Eq. (1) such that the matrix B is a nonsingular $n×n$ matrix. Then the representation of solution Eq. (18) coincides with the conclusion of Theorems 1 and 2 in [12], which leads to the same of the finite time stability results in [27].

5 An Example

Consider the conformable delay differential equations

$(D01.8y)(x)=−By(x−0.5)+f(x),x∈[0,1],ψ(x)=(0.1x2,0.2x)T,ψ′(x)=(0.2x,0.2)T,ψ′′(x)=(0.2,0)T,−0.5≤x≤0,$ (25)

where

$α=1.8,τ=0.5,B = (2002),f(x)=(x1/52x1/5).$

From Theorems 3.1 and 3.2, for all $0≤x≤1$ , and through a basic calculation, we can obtain

$y(x)=(0.025H0.5,1.8(2x1.8)−0.1H0.5,1.8(2x1.8))+(−0.1M0.5,1.8(2x1.8)0.2M0.5,1.8(2x1.8))+(0.2∫−0.50M0.5,1.8(2(x−0.5−υ)1.8)dυ0)+(∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ2∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ)=(y1(x)y2(x)),$

which implies that

$y1(x)=0.025H0.5,1.8(2x1.8)−0.1M0.5,1.8(2x1.8)+0.2∫−0.50M0.5,1.8(2(x−0.5−υ)1.8)dυ+∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

and

$y2(x)=−0.1H0.5,1.8(2x1.8)+0.2M0.5,1.8(2x1.8)+2∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

where

$H0.5,1.8(2x1.8)={1,−0.5≤x<0,1−2518x1.8,0≤x<0.5,1−2518x1.8+6252106(x−0.5)3.6,0.5≤x<1,$

and

$M0.5,1.8(2x1.8)={(x+0.5),−0.5≤x<0,(x+0.5)−2563x2.8,0≤x<0.5,(x+0.5)−2563x2.8+62513041(x−0.5)4.6,0.5≤x<1.$

Thus the explicit solutions of Eq. (25) are

$y1(x)=0.025H0.5,1.8(2x1.8)−0.1M0.5,1.8(2x1.8)+0.2∫−0.5x−0.5M0.5,1.8(2(x−0.5−υ)1.8)dυ+0.2∫x−0.50M0.5,1.8(2(x−0.5−υ)1.8)dυ+∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

$y2(x)=−0.1H0.5,1.8(2x1.8)+0.2M0.5,1.8(2x1.8)+2∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

where $0≤x≤0.5$ , which implies that

$y1(x)=−251197x3.8+5126x2.8+12x2−5144x1.8,$

$y2(x)=−563x2.8+x2+536x1.8+15x,$

and

$y1(x)=0.025H0.5,1.8(2x1.8)−0.1M0.5,1.8(2x1.8)+0.2∫−0.5x−1M0.5,1.8(2(x−0.5−υ)1.8)dυ+0.2∫x−10M0.5,1.8(2(x−0.5−υ)1.8)dυ+∫0x−0.5M0.5,1.8(2(x−0.5−υ)1.8)dυ+∫x−0.5xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

$y2(x)=−0.1H0.5,1.8(2x1.8)+0.2M0.5,1.8(2x1.8)+2∫0x−0.5M0.5,1.8(2(x−0.5−υ)1.8)dυ+2∫x−0.5xM0.5,1.8(2(x−0.5−υ)1.8)dυ,$

where $0.5≤x≤1$ , which implies that

$y1(x)=625365148(x−0.5)5.6−12526082(x−0.5)4.6−1001197(x−0.5)3.8+12516848(x−0.5)3.6−251197x3.8+5126x2.8+12x2−5144x1.8,$

$y2(x)=12513041(x−0.5)4.6−2501197(x−0.5)3.8−1254212(x−0.5)3.6−563x2.8+x2+536x1.8+15x.$

By calculating we obtain $η=max{‖ψ‖C,‖ψ′‖C,‖ψ′′‖C}=0.3$ , $‖B‖=2$ , $‖f‖C=3$ , $Eα(20.8,L)=4.0104$ , $Eα(22.8,L+0.5)=2.278$ , $Eα(22.8,L)=1.4871$ , then we set $δ=0.31>0.3=η$ . Fig. 1 shows the state $y(x)$ and the norm $‖y(x)‖$ of Eq. (25). Now Theorem 4.1 implies that $‖y(x)‖≤5.930254$ , we just take $β=5.9303$ , which implies that $‖y(x)‖<β$ and Eq. (25) is finite time stable.

images

Figure 1: The state y(x) and ||y(x)|| of Eq. (25)

6 Conclusion

In this work, using new conformable delayed matrix functions, we derived explicit solutions of linear conformable fractional delay systems of order $α∈(1,2]$ , which extend and improve the corresponding and existing ones in [12,13] in the case of $α=2$ without any restrictions on the matrix coefficient of the linear part, by removing the condition that B is a nonsingular matrix and replacing the matrix coefficient of the linear part $B2$ in [12] by an arbitrary, not necessarily squared, matrix. In addition, using the formula of general solutions and a norm estimation of the conformable delayed matrix functions, we established some sufficient conditions for the finite time stability results, which extend and improve the existing ones in [27] in the case of $α=2$ . Ultimately, an illustrative example was given to show the validity of the proposed results.

Following the topic of this paper, we outline some possible next research directions. The first direction will include applying the results of this paper on control problems for conformable fractional delay systems of order $α∈(1,2]$ . The second direction is to consider the explicit solutions of linear conformable fractional delay systems of the form

$D0α(D0αy)(x)=−By(x−τ),forx≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for−τ≤x≤0,0<α≤1,$

which lead to new results on stability and control problems. Depending on these results and delayed arguments, we will try to prove a generalized Lyapunov-type inequality for the conformable and sequential conformable boundary value problems

$(Daαy)(x)=−By(x−τ),forx∈(a,b),α∈(1,2]y(a)≡y(b)=Θ,−τ≤x≤0,$

$(Da2αy)(x)=−By(x−τ),forx∈(a,b),α∈(12,1]y(a)≡y(b)=Θ,−τ≤x≤0,$

and

$Daα1(Daα2y)(x)=−By(x−τ),forx∈(a,b),α1,α2∈(0,1]y(a)≡y(b)=Θfor−τ≤x≤0,1<α1+α2≤2,$

which leads to new results on the conformable Sturm-Liouville eigenvalue problem.

Acknowledgement: The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project No. (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding Statement: Princess Nourah bint Abdulrahman University Researchers Supporting Project No. (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Cite This Article

APA Style

Elshenhab, A.M., Wang, X., Mofarreh, F., Bazighifan, O. (2023). Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay. Computer Modeling in Engineering & Sciences, 134(2), 927–940. https://doi.org/10.32604/cmes.2022.021512

Vancouver Style

Elshenhab AM, Wang X, Mofarreh F, Bazighifan O. Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay. Comput Model Eng Sci. 2023;134(2):927–940. https://doi.org/10.32604/cmes.2022.021512

IEEE Style

A. M. Elshenhab, X. Wang, F. Mofarreh, and O. Bazighifan, “Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay,” Comput. Model. Eng. Sci., vol. 134, no. 2, pp. 927–940, 2023. https://doi.org/10.32604/cmes.2022.021512

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Copyright © 2023 The Author(s). Published by Tech Science Press.
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